Hölder Continuity of Solutions of an Elliptic p(x)-Laplace Equation Uniformly Degenerate on a Part of the Domain

Abstract

In a domain D ⊂ ℝⁿ divided by a hyperplane Σ into two parts D⁽¹⁾ and D⁽²⁾, we consider a p(x)-Laplace type equation with a small parameter and with exponent p(x) that has a logarithmic modulus of continuity in each part of the domain and undergoes a jump on Σ when passing from D⁽²⁾ to D⁽¹⁾. Under the assumption that the equation uniformly degenerates with respect to the small parameter in D⁽¹⁾, we establish the Hölder continuity of solutions with Hölder exponent independent of the parameter.